Call a subset $Z$ of $\mathbb{S}^n$ ambiently-reversible, if there is an orientation-reversing self-homeomorphism $h: \mathbb{S}^n \to \mathbb{S}^n$ fixing $Z$ pointwise.

Question 1: Which subsets of $\mathbb{S}^n, n \geq 2$ are ambiently-reversible?

This may be hard, so let me ask a more specific question:

Question 2: Let $Z \subset \mathbb{S}^2$ be closed. Is it true that $Z$ is ambiently-reversible iff it is homeomorphic with a subspace of $\mathbb{S}^1$?

Note that if $Z$ is contained in a homeomorph $K$ of $\mathbb{S}^1$ where $K\subset \mathbb{S}^2$, then $Z$ is ambiently-reversible. To see this, apply Jordan-Shoenflies to $K$, and reflect its two sides. This can be used, in combination with the Denjoy–Riesz theorem, to show that every closed 0-dimensional subset of $\mathbb{S}^2$ is ambiently-reversible.

Remark: If $Z$ is ambiently-reversible, then so is its closure.

See also this question.

Call a subset $Z$ of $\mathbb{S}^n$ ambiently-reversible, if there is an orientation-reversing self-homeomorphism $h: \mathbb{S}^n \to \mathbb{S}^n$ fixing $Z$ pointwise.

Question 1: Which subsets of $\mathbb{S}^n, n \geq 2$ are ambiently-reversible?

This may be hard, so let me ask a more specific question:

Question 2: Let $Z \subset \mathbb{S}^2$ be closed. Is it true that $Z$ is ambiently-reversible iff it is homeomorphic with a subspace of $\mathbb{S}^1$?

An amusing excercise: show that $\mathbb{Z}^2$ is ambiently-reversible in $\mathbb{R}^2$.

Note that if $Z$ is contained in a homeomorph $K$ of $\mathbb{S}^1$ where $K\subset \mathbb{S}^2$, then $Z$ is ambiently-reversible. To see this, apply Jordan-Shoenflies to $K$, and reflect its two sides. This can be used, in combination with the Denjoy–Riesz theorem, to show that every closed 0-dimensional subset of $\mathbb{S}^2$ is ambiently-reversible. (This solves the above excercise.)

Remark: If $Z$ is ambiently-reversible, then so is its closure.

See also this question.